Some methods for predicting changes in the topography, i.e., the shape of the surface, of a semiconductor workpiece in a manufacturing process, such as etching of the workpiece, deposition of a film on the workpiece, and oxidation of the workpiece, use the modified diffusion model. In these methods, the topography of the workpiece is represented by a surface composed of virtual particles distributed in a space that includes the surface. In these methods, changes in topography are considered to result from the diffusion of the virtual particles. A method employing the modified diffusion model as applied to wet etching is briefly described below.
FIG. 4 illustrates changes in the topography of a semiconductor workpiece during wet etching. A silicon oxide film 21 is disposed on a silicon substrate 20 and a mask 22 is disposed on the oxide film 21. An etchant flows through an opening in the mask 22 and etches the substrate 20 and the oxide film 21. An etching front 2 moves in the directions indicated by the arrows 23. The arrows 24 indicate the flow of the etchant. The portions of the substrate 20 and of the oxide film 21 that have been removed by etching are indicated by reference number 25. The changes in the topography of the materials in the etching process are predicted using the modified diffusion model as described below.
Initially, the concentration of the virtual particles distributed in the space is C(r, t), where r=r (X, Y, Z) is the spatial position specified by the coordinates X, Y, and Z and t is time. An etching front is expressed as a contour surface C(r, t)=C1, where C1 is a constant. Examples of contour surfaces for different constant values of C are shown in FIG. 5. For example, C(r, t)=0.30 is indicated by a dashed line 26. The contour surface representing the etching front, which may be C1=0.50, is shown by the solid line 2. The concentration C(r, t) of the virtual particles represents the topography of the workpiece, is high in the etched portion 25 of the materials shown in FIG. 4, and is low within the substrate 20. When using the modified diffusion model, after the etched topography has been expressed at an initial time as a concentration C(r, t) of virtual particles as described above, the diffusion Equation (1) set out below is solved to determine the concentration C(r, t) at a later time. EQU (.differential.C/.differential.t)=Dx(.differential..sup.2 C/.differential.X.sup.2)+Dy(.differential..sup.2 C/.differential.Y.sup.2)+Dz(.differential..sup.2 C/.differential.Z.sup.2) (1)
Thus, the etching front 2 is determined as the surface where C(r, t)=C1. The diffusion coefficients Dx, Dy, and Dz employed in Equation (1) are determined from actual etching rate measurements. For example, in a case where the etching rate of the substrate 20 is higher than that of the oxide film 21, the diffusion coefficients for the substrate 20 are set to larger values than the diffusion coefficients for the oxide film 21. Because of the different diffusion coefficients, the rate at which the surface of the substrate 20 changes is faster than the rate at which the surface of the oxide film 21 changes. The diffusion coefficients of the etched portion 25 are set to large values compared to those of the substrate 20 and the oxide film 21 so that the concentration of virtual particles in the etched portion 25 is substantially constant.
Prediction of the topography of a semiconductor workpiece using the modified diffusion model has been briefly described. The procedure for solving the diffusion Equation (1) in carrying out the prediction is described next.
Initially, a space including the surface of a semiconductor workpiece is divided into a large number of points P(i, j) in a mesh or grid, as shown in FIG. 6. Next, an array C(i, j) of the virtual particles representing the surface of the workpiece and an array m(i, j) of the various materials of the workpiece is prepared with respect to the individual mesh points P(i, j). If no material that will be etched exists at a mesh point P(i, j), m(i, j) is zero. If the oxide film 21 is present at a mesh point P(i, j), m(i, j) is 1 for that mesh point. If the substrate 20 is present at a mesh point P(i, j), then m(i, j) is 2 at that mesh point. The array m(i, j) of materials is used to select the diffusion coefficients Dx, Dy, and Dz at the individual mesh points. The diffusion Equation (1) is solved using these arrays and diffusion coefficients to determine the concentration distribution of virtual particles at a time t.
The concentrations obtained for the individual mesh points P(i, j) are interpolated to draw a contour surface C=C1, e.g., C1=0.50, to represent an etching front 2, as shown in FIG. 7. In FIG. 7, the numbers written below the respective mesh points P(i, j) represent the concentrations C(i, j) determined for those points.
Predictions of the topographic changes produced by one process, i.e., etching, have been described. In an actual semiconductor manufacturing operation, topographic predictions must be made for a plurality of processes. For example, a contact hole may be formed by two sequential processes, e.g., wet etching followed by dry etching, as indicated in FIGS. 8 and 9, In a first process, indicated in FIG. 8, a relatively wide recess 27 is formed in the oxide film by wet etching in order to improve the coverage of an aluminum metallization subsequently deposited on the surface. In a second, subsequent process illustrated in FIG. 9, a hole 28 extending from the recess 27 to the substrate 20 is formed by dry etching.
In order to predict the topography of a semiconductor workpiece after at least two sequential processes have been carried out, the concentrations C(i, j) as well as the materials m(i, j) at the individual mesh points P(i, j) which are predicted for the conclusion of the first process, e.g., wet etching, are used as the initial conditions for the subsequent process, e.g., dry etching.
Conventionally, of the materials m(i, j) and the concentrations C(i, j) for the individual mesh points P(i, j), as shown in FIGS. 10A and 10B, respectively, which are determined for the completion of a first process, only the materials array m(i, j) is saved, for example, as indicated by the array of FIG. 10C, for use in topography predictions for subsequent processes. In FIG. 10A, the numbers below the respective mesh points P(i, j) represent the materials m(i, j) at the respective mesh points. In FIG. 10B, the numbers written below the individual mesh points represent the concentrations C(i, j) for the respective mesh points. The etching front is indicated by the number 2 and the interface between the substrate 20 and the oxide film 21 is denoted by the number 3.
In the second process, the initial values of the materials m(i, j), as shown in FIG. 10D for the respective mesh points, are the values at the end of the first process as indicated in FIG. 10A. The concentration C(i, j) is 1.0 where no material to be etched exists and 0.0 where material to be etched is present, as shown in FIG. 10E. Use of only the materials array, i.e., m(i, j), without the use of the concentration array C(i, j), decreases the amount of data employed in each subsequent step of a prediction method.
Recent manufacturing techniques for making integrated circuits generally employ at least one hundred sequential processes. A three-dimensional topography prediction employs several million mesh points P. In using the modified diffusion model, the concentrations, other than those for the contour surface where C(r, t)=C1, representing an etching front, do not much affect the predicted topography. Hence, conventionally, only the materials array m(i, j) from one process is used as an initial value for the next subsequent process without a similar use of the concentration array C(i, j) to provide initial conditions for the next subsequent process.
Since the concentration array from the conclusion of the first process is discarded, the initial values of the concentrations C(i, j) for the second process are set to either zero or 1. However, this step prevents the topography predicted at the completion of the first process from being completely and accurately reproduced at the beginning of the second process. As a result, the accuracy of the topography prediction decreases with each subsequent process. When the topography is predicted after one hundred or more sequential processes, the accuracy of the prediction of the final topography is poor.